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arXiv:0711.1621v1 [math.FA] 10 Nov 2007
ARENS REGULARITY OF MODULE ACTIONS AND THE SECOND
ADJOINT OF A DERIVATION
S. MOHAMMADZADEH AND H. R. E. VISHKI
Abstract. In this paper, ﬁrst we give a simple criterion for the Arens regularity of
a bilinear mapping on normed spaces, which applies in particular to Banach mo dule
actions and then we investigate those conditions under which the second adjoint of a
derivation into a dual Banach module is again a derivation. As a consequence of the
main result, a simple and direct proof for several older results is also included.
1. Introduction
In his pioneering paper, [3], Arens has shown that a bounded bilinear map f : X×Y −→
Z on normed spaces, has two natural but, in general, diﬀerent extensions to the bilinear
maps f
∗∗∗
and f
r∗∗∗r
. When these extensions are equal, f is said to be (Arens) regular.
If the multiplication of a Banach algebra A enjoys this property, then A itself is called
(Arens) regular.
In this paper we ﬁrst provide a criterion for the regularity of a bounded bilinear map
(Theorem 2.1 below), by showing that f is regular if and only if f
∗∗∗∗
(Z
∗
, X
∗∗
) ⊆ Y
∗
;
which in turn covers some older results of [10], [5] on this topic (see Corollaries 2.2 and 2.3
below). Then we apply the above mentioned criterion for the module actions of a Bana ch
A−module X, which in turn give rise to the module actions of A
∗∗
(equipped with each
of the Arens products) on X
∗∗
and X
∗∗∗
, in a natural way. In this direction we present
Propositions 3 .1 , 3.3 and 3.6, which generalize some results of [4], [6], [10], and [11].
For a Banach A−module X, the second adjoint D
∗∗
: A
∗∗
→ X
∗∗∗
of a derivation
D : A → X
∗
is trivially a linear extension of D. A problem which is of interest is under
2000 Mathematics Subject Classiﬁcation. 46H20, 46H25.
Key words and phrases. Arens product, bounded bilinear map, Banach module action, derivation,
module action, second dual.
1
2 S. MOHAMMADZADEH AND H. R. E. VISHKI
what conditions D
∗∗
is again a derivation. Dales, RodriguezPalacios and Velasco in [10]
studied this problem for the special case X = A, and they showed that D
∗∗
is a derivation
if and only if D
∗∗
(A
∗∗
) · A
∗∗
⊆ A
∗
, (see [10, Theorem 7.1]). We extend t heir result for a
general derivation D : A → X
∗
with a direct proof (see Theorem 4.2 below). This theorem
in turn extends some o t her results of [10] and [6] for a general derivation D : A → X
∗
.
For terminology and background materials we follow [8], as far as possible.
2. Arens regularity of bilinear maps
For a normed space X, we denote by X
∗
the topological dual of X. We write X
∗∗
for
(X
∗
)
∗
, and so on. Throughout the paper, we usually identify a normed space with its
canonical image in its second dual.
Let X, Y and Z be normed spaces and let f : X × Y −→ Z be a bounded bilinear
map. The adjoint f
∗
: Z
∗
× X −→ Y
∗
of f is deﬁned by
hf
∗
(z
∗
, x), yi = hz
∗
, f(x, y)i (x ∈ X, y ∈ Y, z
∗
∈ Z
∗
),
which is also a bounded bilinear map. By setting f
∗∗
= (f
∗
)
∗
and continuing this way, the
mappings f
∗∗
: Y
∗∗
×Z
∗
−→ X
∗
, f
∗∗∗
: X
∗∗
×Y
∗∗
−→ Z
∗∗
and f
∗∗∗∗
: Z
∗∗∗
×X
∗∗
−→ Y
∗∗∗
may be deﬁned similarly.
The mapping f
∗∗∗
is the unique extension of f such that f
∗∗∗
(·, y
∗∗
) is w
∗
−continuous
for every y
∗∗
∈ Y
∗∗
. Also f
∗∗∗
(x, ·) is w
∗
−continuous for every x ∈ X.
We also denote by f
r
the ﬂip map of f, that is the bounded bilinear map f
r
: Y ×X −→
Z deﬁned by f
r
(y, x) = f(x, y) (x ∈ X, y ∈ Y ). It may be raised as above a bounded
bilinear map f
r∗∗∗r
: X
∗∗
×Y
∗∗
−→ Z
∗∗
which in turn is a unique extension of f such that
f
r∗∗∗r
(x
∗∗
, ·) is w
∗
−continuous for every x
∗∗
∈ X
∗∗
, and also f
r∗∗∗r
(·, y) is w
∗
−continuous
for every y ∈ Y . One may also easily verify that
f
∗∗∗
(x
∗∗
, y
∗∗
) = w
∗
− lim
α
lim
β
f(x
α
, y
β
)
and
f
r∗∗∗r
(x
∗∗
, y
∗∗
) = w
∗
− lim
β
lim
α
f(x
α
, y
β
),
ARENS REGULARITY OF BILINEAR MAPPINGS · · · 3
where {x
α
} and {y
β
} are nets in X and Y which converge to x
∗∗
and y
∗∗
in the w
∗
−topologies,
respectively.
The map f is called Arens regular when the equality f
∗∗∗
= f
r∗∗∗r
holds.
It is easy to verify that, for the multiplication map π : A×A −→ A of a Banach algebra
A, π
∗∗∗
and π
r∗∗∗r
are actually the socalled ﬁrst and second Arens products [3], which will
be denoted by and ♦, respectively. The Banach algebra A is said to be Arens regular
if the multipliction map π is Arens regular, or equivalently = ♦ on the whole of A
∗∗
.
This is the case, for example, for all C
∗
−algebras, [7], and also for L
1
(G) if (and only if)
G is ﬁnite, [12]. The interested reader may refer to [9] for ample information about the
Arens r egularity problem on a wide variety of Banach algebras.
We commence with the main theorem of this section which provide a criterion concern
ing t o the regularity of a bounded bilinear map.
Theorem 2.1. For a bounded bilinear map f : X × Y −→ Z the following statements
are equivalent:
(i) f is regular.
(ii) f
∗∗∗∗
= f
r∗∗∗∗∗r
.
(iii) f
∗∗∗∗
(Z
∗
, X
∗∗
) ⊆ Y
∗
.
(iv) The linear map x 7−→ f
∗
(z
∗
, x) : X −→ Y
∗
is weakly compact for every z
∗
∈ Z
∗
.
Proof. Let x
∗∗
∈ X
∗∗
, y
∗∗
∈ Y
∗∗
, z
∗∗∗
∈ Z
∗∗∗
and z
∗
∈ Z
∗
be arbitrary. If (i) holds then
hf
∗∗∗∗
(z
∗∗∗
, x
∗∗
), y
∗∗
i = hz
∗∗∗
, f
∗∗∗
(x
∗∗
, y
∗∗
)i
= hz
∗∗∗
, f
r∗∗∗r
(x
∗∗
, y
∗∗
)i
= hz
∗∗∗
, f
r∗∗∗
(y
∗∗
, x
∗∗
)i
= hx
∗∗
, f
r∗∗∗∗
(z
∗∗∗
, y
∗∗
)i
= hf
r∗∗∗∗∗
(x
∗∗
, z
∗∗∗
), y
∗∗
i.
Therefore f
∗∗∗∗
= f
r∗∗∗∗∗r
, as claimed.
4 S. MOHAMMADZADEH AND H. R. E. VISHKI
The implication (ii) ⇒ (iii) follows trivially from the fact that
f
∗∗∗∗
(z
∗
, x
∗∗
) = f
r∗∗∗∗∗
(x
∗∗
, z
∗
) = f
r∗∗∗∗∗

X
∗∗
×Z
∗
(x
∗∗
, z
∗
) = f
r∗∗
(x
∗∗
, z
∗
) ∈ Y
∗
.
That (iii) is equivalent to (iv) is o bvious; indeed, if we denote f
∗
(z
∗
, ·) by L, then it is
easy to show that L
∗∗
= f
∗∗∗∗
(z
∗
, ·). Now the conclusion follows from the fact that L is
weakly compact if and only if L
∗∗
(X
∗∗
) ⊆ Y
∗
.
For (iii) ⇒ (i), suppose that f
∗∗∗∗
(Z
∗
, X
∗∗
) ⊆ Y
∗
and let {x
α
}, {y
β
} be two nets in X
and Y that converge to x
∗∗
, y
∗∗
in the w
∗
−topologies, respectively; then
hf
∗∗∗
(x
∗∗
, y
∗∗
), z
∗
i = hf
∗∗∗∗
(z
∗
, x
∗∗
), y
∗∗
i
= lim
β
hf
∗∗∗∗
(z
∗
, x
∗∗
), y
β
i
= lim
β
hf
∗∗∗
(x
∗∗
, y
β
), z
∗
i
= lim
β
hx
∗∗
, f
∗∗
(y
β
, z
∗
)i
= lim
β
lim
α
hf
∗∗
(y
β
, z
∗
), x
α
i
= lim
β
lim
α
hz
∗
, f(x
α
, y
β
)i
= hf
r∗∗∗r
(x
∗∗
, y
∗∗
), z
∗
i.
It f ollows that f is regular and this completes the proof.
As the ﬁrst application of Theorem 2.1, we may present the following results of [10],
with a direct proof.
Corollary 2.2 ([10, Propositions 4.1 and 4.4]). For a bounded bilinear map f : X ×Y −→
Z, the following statements are equivalent:
(i) f and f
r∗
are regular.
(ii) f
∗∗∗r∗r
= f
r∗r∗∗∗
.
(iii) f
∗∗∗∗
(Z
∗∗∗
, X
∗∗
) ⊆ Y
∗
.
Proof. The implication (i) ⇒ (ii) follows trivially.
ARENS REGULARITY OF BILINEAR MAPPINGS · · · 5
If (ii) holds then f
∗∗∗∗
= f
r∗r∗∗
. Indeed, for every x
∗∗
∈ X
∗∗
, y
∗∗
∈ Y
∗∗
and z
∗∗∗
∈ Z
∗∗∗
,
hf
∗∗∗∗
(z
∗∗∗
, x
∗∗
), y
∗∗
i = hf
∗∗∗r∗r
(y
∗∗
, z
∗∗∗
), x
∗∗
i
= hf
r∗r∗∗∗
(y
∗∗
, z
∗∗∗
), x
∗∗
i
= hf
r∗r∗∗
(z
∗∗∗
, x
∗∗
), y
∗∗
i.
As f
r∗r∗∗
(Z
∗∗∗
, X
∗∗
) always lies in Y
∗
, we have reached (iii).
For (iii) ⇒ (i), since f
∗∗∗∗
(Z
∗
, X
∗∗
) ⊆ f
∗∗∗∗
(Z
∗∗∗
, X
∗∗
) ⊆ Y
∗
, Theorem 2.1 implies the
regularity of f, or equivalently f
r∗∗∗∗∗
= f
∗∗∗∗r
, from which
(f
r∗
)
∗∗∗∗
(X
∗∗
, Z
∗∗∗
) = f
r∗∗∗∗∗
(X
∗∗
, Z
∗∗∗
) = f
∗∗∗∗
(Z
∗∗∗
, X
∗∗
) ⊆ Y
∗
.
Now the regularity of f
r∗
follows trivially again by Theorem 2.1.
The main result of Arikan’s paper, ([5, Theorem 2]), is a criterion for the Arens r egular
ity of bilinear mappings which applied it to give a train of results on the regularity of the
algebra l
1
with pointwise multiplication, the algebra L
∞
(G) with convolution, where G is
a compact group, and the traceclass algebra. Now, we give her result as a consequence
of Corolla ry 2.2. It is worthwhile discussing that the assumption that she applied in [5,
Theorem 2] has actually more byproducts; indeed, as we shall see in t he next corollary,
both f and f
r∗
are regular, however she deduces merely the regularity of f.
Corollary 2.3 (see [5, Theorem 2]). Let f : X × Y −→ Z and g : X × W −→ Z be
bounded bilinear mappings and let h : Y −→ W be a bounded linear mapping such that
f(x, y) = g(x, h(y)), for all x ∈ X, y ∈ Y . If h is weakly compact, then both f and f
r∗
are regular.
Proof. Using the equality f (x, y) = g(x, h(y)), a standard argument applies to show that
f
∗∗∗∗
= h
∗∗∗
◦ g
∗∗∗∗
. The weak compactness of h implies that of h
∗
, from which we have
h
∗∗∗
(W
∗∗∗
) ⊆ Y
∗
. Therefore
f
∗∗∗∗
(Z
∗∗∗
, X
∗∗
) = h
∗∗∗
(g
∗∗∗∗
(Z
∗∗∗
, X
∗∗
)) = h
∗∗∗
(W
∗∗∗
) ⊆ Y
∗
.
6 S. MOHAMMADZADEH AND H. R. E. VISHKI
Now Corollary 2.2 implies that both f and f
r∗
are regular, as claimed.
3. Arens regularity of module actions
Let A be a Banach algebra, X be a Banach space and let π
1
: A × X −→ X be a
bounded bilinear map. Then the pair (π
1
, X) is said to be a left Banach A−module if
π
1
(ab, x) = π
1
(a, π
1
(b, x)), for every a, b ∈ A, x ∈ X. A right Banach A−module (X , π
2
)
may be deﬁned similarly. A triple (π
1
, X, π
2
) is said to be a Banach A−module if (π
1
, X)
and (X, π
2
) are left and right Banach A−modules, respectively, and for every a, b ∈ A, x ∈
X,
π
1
(a, π
2
(x, b)) = π
2
(π
1
(a, x), b).
If so, then trivially (π
2
r∗r
, X
∗
, π
∗
1
) is the dual Banach A−module of (π
1
, X, π
2
). Also
(π
∗∗∗
1
, X
∗∗
, π
∗∗∗
2
) is a Banach (A
∗∗
, )−module as well as (π
r∗∗∗r
1
, X
∗∗
, π
r∗∗∗r
2
) is a Banach
(A
∗∗
, ♦)−module. To verify these, one may carefully check that the various associativity
rules follow from the analog ous rules that show that (π
1
, X, π
2
) is a Banach A−module.
Now if we continue dualizing we shall reach (π
∗∗∗r∗r
2
, X
∗∗∗
, π
∗∗∗∗
1
) as the dual Banach
(A
∗∗
, )−module of (π
∗∗∗
1
, X
∗∗
, π
∗∗∗
2
) as well as (π
r∗∗∗∗r
2
, X
∗∗∗
, π
r∗∗∗r∗
1
) as the dual Banach
(A
∗∗
, ♦)−module of (π
r∗∗∗r
1
, X
∗∗
, π
r∗∗∗r
2
).
From now on, for a Banach A−module X we usually use the above mentioned canonical
module actions on X
∗
, X
∗∗
and X
∗∗∗
, unless otherwise stipulated explicitly. However, fo r
brevity of notation, when there is no risk of confusion, we may use them without the
speciﬁed module actions.
According to the inclusions
π
∗∗∗r∗r
2
(A
∗∗
, X
∗
) = π
∗∗
2
(A
∗∗
, X
∗
) ⊆ X
∗
and
π
r∗∗∗r∗
1
(X
∗
, A
∗∗
) = π
r∗∗r
1
(X
∗
, A
∗∗
) ⊆ X
∗
,
ARENS REGULARITY OF BILINEAR MAPPINGS · · · 7
we deduce that (π
∗∗
2
, X
∗
) is a left (A
∗∗
, )−submodule of X
∗∗∗
, while (X
∗
, π
r∗∗r
1
) is a right
(A
∗∗
, ♦)−submodule of X
∗∗∗
. The next result determines when (π
∗∗
2
, X
∗
, π
r∗∗r
1
) is actually
a (A
∗∗
, ) (respectively, (A
∗∗
, ♦))−submodule o f X
∗∗∗
.
Proposition 3.1. Let (π
1
, X, π
2
) be a Banach A−module. Then
(i) X
∗
is a (A
∗∗
, )−submodule of X
∗∗∗
if and only if π
1
is regular.
(ii) X
∗
is a (A
∗∗
, ♦)−submodule of X
∗∗∗
if and only if π
2
is regular.
Proof. We prove only (i), the other one uses the same argument. It is trivial tha t X
∗
is a
right (A
∗∗
, )−submodule of X
∗∗∗
if and only if π
∗∗∗∗
1
(X
∗
, A
∗∗
) ⊆ X
∗
and by Theorem 2.1
this is nothing but the regularity of π
1
; which establishes (i).
As a rapid consequence of the latter result, we examine it for π
1
= π and π
2
= π
r
on
X = A and we thus have the following result of Dales, RodriguezPalacios and Velasco,
[10].
Corollary 3.2 ([10, Proposition 5.2]). Fo r a Banach alg e bra A the following statements
are equivalent:
(i) A is Arens regular.
(ii) A
∗
is a (A
∗∗
, )−submodule of A
∗∗∗
.
(iii) A
∗
is a (A
∗∗
, ♦)−submodule of A
∗∗∗
.
Let (π
1
, X, π
2
) be a Banach A−module. As we mentioned just before Proposition 3.1,
(π
∗∗
2
, X
∗
) and (X
∗
, π
r∗∗r
1
) are left (A
∗∗
, )−submodule and right (A
∗∗
, ♦)−submodule of
X
∗∗∗
, respectively. So if we assume that A is Arens regular then (π
∗∗
2
, X
∗
) and (X
∗
, π
r∗∗r
1
)
are left and right A
∗∗
−modules, resp ectively. The next result deals with the question
when (π
∗∗
2
, X
∗
, π
r∗∗r
1
) is actually a A
∗∗
−module.
Proposition 3.3. Let A be Arens regular and let (π
1
, X, π
2
) be a Banach A−module.
Then (π
∗∗
2
, X
∗
, π
r∗∗r
1
) is a B anach A
∗∗
−module if and only if the bilinear map
θ
x
: A × A −→ X, θ
x
(a, b) = π
1
(a, π
2
(x, b)) = π
2
(π
1
(a, x), b) (a, b ∈ A)
is regular, fo r all x ∈ X.
8 S. MOHAMMADZADEH AND H. R. E. VISHKI
Proof. The triple (π
∗∗
2
, X
∗
, π
r∗∗r
1
) is a Banach A
∗∗
−module if and only if for all a
∗∗
, b
∗∗
∈
A
∗∗
, x
∗
∈ X
∗
and x ∈ X,
hπ
1
r∗∗r
(π
2
∗∗
(b
∗∗
, x
∗
), a
∗∗
), xi = hπ
2
∗∗
(b
∗∗
, π
1
r∗∗r
(x
∗
, a
∗∗
)), xi.
Let {a
α
} and {b
β
} be two nets in A that converge to a
∗∗
and b
∗∗
, respectively, in t he
w
∗
−topology of A
∗∗
. Then a direct veriﬁcation reveals that
hπ
1
r∗∗r
(π
2
∗∗
(b
∗∗
, x
∗
), a
∗∗
), xi = lim
α
lim
β
hx
∗
, π
2
(π
1
(a
α
, x), b
β
)i = lim
α
lim
β
hx
∗
, θ
x
(a
α
, b
β
)i
and
hπ
2
∗∗
(b
∗∗
, π
1
r∗∗r
(x
∗
, a
∗∗
)), xi = lim
β
lim
α
hx
∗
, π
1
(a
α
, π
2
(x, b
β
)) = lim
β
lim
α
hx
∗
, θ
x
(a
α
, b
β
)i.
Thus (π
∗∗
2
, X
∗
, π
r∗∗r
1
) is a Banach A
∗∗
−module if and only if θ
x
is regular, for all x ∈ X.
As a consequence, we give the next result of Bunce and Paschke, [6].
Corollary 3.4 ([6, Proposition 1.1]). Let A be a C
∗
−algebra and let (π
1
, X, π
2
) be a
Banach A−module. Then (π
∗∗
2
, X
∗
, π
r∗∗r
1
) is a B anach A
∗∗
−module.
Proof. By Proposition 3.3 it is enough to show that θ
x
: A×A −→ X is regular, for all x ∈
X. Applying Theorem 2.1, the regularity of θ
x
is equivalent to the weak compactness of
the linear mapping a 7→ θ
x
∗
(x
∗
, a) : A −→ A
∗
, which is guaranteed by the f act that, every
bounded linear map fro m a C
∗
−algebra to the predual of a W
∗
− algebra is automatically
weakly compact (see [1, Corollary II 9 ]).
Remarks 3.5.
(i) If either π
1
or π
2
is regular then trivially θ
x
is regular, for each x ∈ X. Therefore,
in the Arens regular setting for A, using Proposition 3.3, X
∗
is a A
∗∗
−module, which is
actually a A
∗∗
−submodule of X
∗∗∗
, (see Proposition 3.1).
(ii) In the Arens regular setting for A, if θ
x
is regular for each x ∈ X then (π
∗∗
2
, X
∗
, π
r∗∗r
1
)
is a Banach A
∗∗
−module by Proposition 3.3, which gives rise naturally the dual Ba
nach A
∗∗
−module (π
r∗∗∗r
1
, X
∗∗
, π
∗∗∗
2
). On the other hand we have the canonical Banach
ARENS REGULARITY OF BILINEAR MAPPINGS · · · 9
A
∗∗
−modules (π
∗∗∗
1
, X
∗∗
, π
∗∗∗
2
) and (π
r∗∗∗r
1
, X
∗∗
, π
r∗∗∗r
2
). It is worthwhile to note that the
involved dual module does not coincide with the canonical modules, in general. However,
it is obvious that it coincides with one of them if (and only if) either π
1
or π
2
is regular.
It is shown in [6, Propo sition 1.2] that the involved module actions coincide when A is a
C
∗
−algebra and X
∗
is weakly sequentially complete; however, according to what we shall
demonstrate in the following, under these conditions on A and X the module actions π
1
and π
2
are r egular.
(iii) Let (π
1
, X, π
2
) be a Banach A−module, if A is a C
∗
−algebra and X
∗
is weakly
sequentially complete, then by [2, Theorem 4.2] the bounded linear mappings a 7−→
π
∗
1
(x
∗
, a) and a 7−→ π
r∗
2
(x
∗
, a) f r om A to X
∗
are weakly compact for all x
∗
∈ X
∗
. Now
applying Theorem 2.1 for π
1
and π
r
2
shows that π
1
and π
2
are regular. Note that the weak
sequential completeness of X
∗
is essential and can not be removed in general. For instance,
let A be the C
∗
−algebra of compact o perators on a separable, inﬁnite dimensional Hilbert
space H and let X be the traceclass operators on H; then a direct veriﬁcation reveals that
the usual A−module action on X is not regular (see the example just before Propo sition
2.1 in [6]).
Dales, RodriguezPalacios and Velasco in [10, Proposition 4.5] (see also, [4, Theorem
4] and [11, Theorem 3 .1]) have shown that, if A is Arens regular with a bounded left
approximate identity, then π
r∗
: A
∗
× A −→ A
∗
is regular if and only if A is reﬂexive
as a Banach space. (Note that, the equality π
∗∗∗r∗r
= π
r∗r∗∗∗
which they used in [10,
Proposition 4.5] is equivalent to the regularity of both A and π
r∗
, see Corollary 2.2). In
the next propo sition we generalize their result, which also shows t hat the hypothesis of
Arens regularity of A in [10, Proposition 4.5] is superﬂuous. Before proceeding, we recall
that, when (π
1
, X) is a left Banach A−module then a bounded net {e
α
} in A is said to be
a left approximate identity for X, if π
1
(e
α
, x) −→ x, for each x ∈ X. As a consequence
of the socalled Cohen Factorization Theorem, see [8], it is known that a bounded left
approximate identity of A is that of X if and only if π
1
(A, X) = X. The same situation
happens for the right Banach A−module (X, π
2
).
10 S. MOHAMMADZADEH AND H. R. E. VISHKI
Proposition 3.6. Let (π
1
, X) and (X, π
2
) be left and righ t Banach A−modules respec
tively.
(i) If A has a bounded left approximate identity for X, then π
r∗
1
is regular if and onl y
if X is reﬂexive.
(ii) I f A has a bounded right approximate i dentity for X, then π
∗
2
is regular if and only
if X is reﬂexive.
Proof. (i) If X is reﬂexive, then
(π
r∗
1
)
∗∗∗∗
(A
∗∗
, X
∗∗∗
) = π
r∗∗∗∗∗
1
(A
∗∗
, X
∗
) = π
r∗∗
1
(A
∗∗
, X
∗
) ⊆ X
∗
;
now Theorem 2.1 implies that π
r∗
1
is regular. For the converse, ﬁrst note that, if {e
α
} is a
bounded left a pproximate identity for X ( in A), then a direct veriﬁcation reveals that for
every x
∗∗
∈ X
∗∗
, π
r∗∗∗
1
(x
∗∗
, e
∗∗
) = x
∗∗
, in which e
∗∗
is a w
∗
−cluster point of {e
α
} in A
∗∗
.
The latter identity in turn implies that x
∗∗∗
= π
r∗∗∗∗∗
1
(e
∗∗
, x
∗∗∗
), for every x
∗∗∗
∈ X
∗∗∗
.
Now the regularity of π
r∗
1
(again by Theorem 2.1) shows that π
r∗∗∗∗∗
1
(e
∗∗
, x
∗∗∗
) ∈ X
∗
. We
thus have x
∗∗∗
∈ X
∗
, or equivalently X is reﬂexive. The proof of (ii) is very similar to
that of (i).
4. The second adjoint of a derivation
Let (π
1
, X, π
2
) be a Banach A−module. A bounded linear mapping D : A −→ X
∗
is
said to be a derivation if D(ab) = D(a) · b + a · D(b), for each a, b ∈ A, or equiva lently
D(ab) = π
∗
1
(D(a), b) + π
r∗r
2
(a, D(b)).
In this section we deal with the question of when the second a djo int D
∗∗
: A
∗∗
−→ X
∗∗∗
of D : A −→ X
∗
is again a derivation. We recall from the beginning of the last section that
(π
∗∗∗r∗r
2
, X
∗∗∗
, π
∗∗∗∗
1
) and (π
r∗∗∗∗r
2
, X
∗∗∗
, π
r∗∗∗r∗
1
) are our canonical Ba na ch (A
∗∗
, )−module
and (A
∗∗
, ♦)− mo dule, respectively. Hence, D
∗∗
: (A
∗∗
, ) −→ X
∗∗∗
is a derivation if and
only if for every a
∗∗
, b
∗∗
∈ A
∗∗
,
D
∗∗
(a
∗∗
b
∗∗
) = π
∗∗∗∗
1
(D
∗∗
(a
∗∗
), b
∗∗
) + π
∗∗∗r∗r
2
(a
∗∗
, D
∗∗
(b
∗∗
)).
ARENS REGULARITY OF BILINEAR MAPPINGS · · · 11
Similarly, D
∗∗
: (A
∗∗
, ♦) −→ X
∗∗∗
is a derivation if and only if
D
∗∗
(a
∗∗
♦ b
∗∗
) = π
r∗∗∗r∗
1
(D
∗∗
(a
∗∗
), b
∗∗
) + π
r∗∗∗∗r
2
(a
∗∗
, D
∗∗
(b
∗∗
)).
We commence with the following lemma.
Lemma 4.1. Let (π
1
, X, π
2
) be a Banac h A−module, and let D : A −→ X
∗
be a de riva
tion. Then fo r all a
∗∗
, b
∗∗
∈ A
∗∗
,
(i) D
∗∗
(a
∗∗
b
∗∗
) = π
∗∗∗∗
1
(D
∗∗
(a
∗∗
), b
∗∗
) + π
r∗r∗∗∗
2
(a
∗∗
, D
∗∗
(b
∗∗
)) and
(ii) D
∗∗
(a
∗∗
♦ b
∗∗
) = π
∗r∗∗∗r
1
(D
∗∗
(a
∗∗
), b
∗∗
) + π
r∗∗∗∗r
2
(a
∗∗
, D
∗∗
(b
∗∗
)).
Proof. (i) Let {a
α
} and {b
β
} be two nets in A that converge to a
∗∗
and b
∗∗
in the
w
∗
−topology of A
∗∗
, respectively. Then for each x
∗∗
∈ X
∗∗
we have
hD
∗∗
(a
∗∗
b
∗∗
), x
∗∗
i = lim
α
lim
β
hD(a
α
b
β
), x
∗∗
i
= lim
α
lim
β
hπ
∗
1
(D(a
α
), b
β
) + π
r∗r
2
(a
α
, D(b
β
)), x
∗∗
i
= lim
α
lim
β
hπ
∗∗
1
(x
∗∗
, D(a
α
)), b
β
i + lim
α
lim
β
hπ
r∗r∗
2
(x
∗∗
, a
α
), D(b
β
)i
= lim
α
hπ
∗∗
1
(x
∗∗
, D(a
α
)), b
∗∗
i + lim
α
hπ
r∗r∗
2
(x
∗∗
, a
α
), D
∗∗
(b
∗∗
)i
= hπ
∗∗∗
1
(b
∗∗
, x
∗∗
), D
∗∗
(a
∗∗
)i + hπ
r∗r∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
), a
∗∗
i
= hπ
∗∗∗∗
1
(D
∗∗
(a
∗∗
), b
∗∗
) + π
r∗r∗∗∗
2
(a
∗∗
, D
∗∗
(b
∗∗
)), x
∗∗
i.
Hence D
∗∗
(a
∗∗
b
∗∗
) = π
∗∗∗∗
1
(D
∗∗
(a
∗∗
), b
∗∗
) + π
r∗r∗∗∗
2
(a
∗∗
, D
∗∗
(b
∗∗
)). A similar argument
applies f or (ii).
Dales, RodriguezPalacios and Velasco, in the main theorem of their paper, [10, The
orem 7.1], have shown t hat the second adjoint D
∗∗
: (A
∗∗
, ) −→ A
∗∗∗
of the deriva
tion D : A −→ A
∗
is a derivation if and only if D
∗∗
(A
∗∗
) · A
∗∗
⊆ A
∗
, or equiva lently
π
r∗∗∗∗
(D
∗∗
(A
∗∗
), A
∗∗
) ⊆ A
∗
, where π is the multiplication of A. The next theorem ex
tends it with a direct proof.
Theorem 4.2. Let (π
1
, X, π
2
) be a Banac h A−module and let D : A −→ X
∗
be a deriva
tion.
12 S. MOHAMMADZADEH AND H. R. E. VISHKI
(i) D
∗∗
: (A
∗∗
, ) −→ X
∗∗∗
is a derivation if and only if π
∗∗∗∗
2
(D
∗∗
(A
∗∗
), X
∗∗
) ⊆ A
∗
.
(ii) D
∗∗
: (A
∗∗
, ♦) −→ X
∗∗∗
is a derivation if and only if π
r∗∗∗∗
1
(D
∗∗
(A
∗∗
), X
∗∗
) ⊆ A
∗
.
Proof. We only prove (i). Let a
∗∗
, b
∗∗
∈ A
∗∗
and x
∗∗
∈ X
∗∗
be arbitrary. Applying
Lemma 4.1 and the equations just before it, one may deduce that D
∗∗
: (A
∗∗
, ) −→ X
∗∗∗
is a derivation if and only if
hπ
∗∗∗r∗r
2
(a
∗∗
, D
∗∗
(b
∗∗
)), x
∗∗
i = hπ
r∗r∗∗∗
2
(a
∗∗
, D
∗∗
(b
∗∗
)), x
∗∗
i;
which holds if a nd only if
hπ
∗∗∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
), a
∗∗
i = hπ
r∗r∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
), a
∗∗
i,
or equivalently,
π
∗∗∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
) = π
r∗r∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
).
It should be noted that π
∗∗∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
) ∈ A
∗∗∗
, while π
r∗r∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
) ∈ A
∗
and
also π
∗∗∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
)

A
= π
r∗r∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
). Thus D
∗∗
is a derivation if and only if
π
∗∗∗∗
2
(D
∗∗
(b
∗∗
), x
∗∗
) ∈ A
∗
.
As immediate consequences of the Theorem 4.2 we have the next corollaries.
Corollary 4.3. Let (π
1
, X, π
2
) be a Banach A−module and let D : A −→ X
∗
be a
derivation. If A is Arens regular then the following statements are equivalent:
(i) D
∗∗
: A
∗∗
−→ X
∗∗∗
is a derivation.
(ii) π
∗∗∗∗
2
(D
∗∗
(A
∗∗
), X
∗∗
) ⊆ A
∗
.
(iii) π
r∗∗∗∗
1
(D
∗∗
(A
∗∗
), X
∗∗
) ⊆ A
∗
.
Corollary 4.4. Let (π
1
, X, π
2
) be a Banach A−module, and let D : A −→ X
∗
be a
derivation.
(i) If both π
2
and π
r∗
2
are Arens regular then D
∗∗
: (A
∗∗
, ) −→ X
∗∗∗
is a derivation.
(ii) If both π
1
and π
∗
1
are Arens regular then D
∗∗
: (A
∗∗
, ♦) −→ X
∗∗∗
is a derivation.
ARENS REGULARITY OF BILINEAR MAPPINGS · · · 13
Proof. (i) If both π
2
and π
r∗
2
are r egular then π
∗∗∗∗
2
(X
∗∗∗
, X
∗∗
) ⊆ A
∗
by Corollary 2.2. In
particular, π
∗∗∗∗
2
(D
∗∗
(A
∗∗
), X
∗∗
) ⊆ A
∗
and the conclusion follows from Theorem 4.2. A
similar proof applies for (ii).
However the regularity of both π
2
and π
r∗
2
in (i) (in turn π
1
and π
∗
1
in (ii)) of the latter
corollary ensure that D
∗∗
is a derivation, but it seems that these conditions impose rather
a strong requirement on the module actions. As another application of Theorem 4.2,
we give the following generalization of [10, Corollary 7.2(i)] which share the imposed
requirements on both module action and the derivation, as well.
Corollary 4.5. Let (π
1
, X, π
2
) be a Banac h A−module and let D : A −→ X
∗
be a weakly
compact derivation.
(i) If π
2
is regular then D
∗∗
: (A
∗∗
, ) −→ X
∗∗∗
is a derivation.
(ii) If π
1
is regular then D
∗∗
: (A
∗∗
, ♦) −→ X
∗∗∗
is a derivation.
Proof. (i) Since D : A −→ X
∗
is weakly compact, D
∗∗
(A
∗∗
) ⊆ X
∗
. On the other hand
the regularity of π
2
implies that π
∗∗∗∗
2
(X
∗
, X
∗∗
) ⊆ A
∗
(see Theorem 2.1). Therefore
π
∗∗∗∗
2
(D
∗∗
(A
∗∗
), X
∗∗
) ⊆ A
∗
, which means that D
∗∗
: (A
∗∗
, ) −→ X
∗∗∗
is a derivation.
We conclude this section with some observations on inner derivations. A linear mapping
D : A −→ X
∗
is said to be an inner derivation if D(a) = x
∗
· a − a · x
∗
for some x
∗
∈ X
∗
,
or equivalently D(a) = π
∗
1
(x
∗
, a) − π
r∗r
2
(a, x
∗
), for all a ∈ A. Then it follows trivially t hat
for every a
∗∗
∈ A
∗∗
,
D
∗∗
(a
∗∗
) = π
∗∗∗∗
1
(x
∗
, a
∗∗
) − π
r∗∗∗∗
2
(x
∗
, a
∗∗
).
If, in addition, we assume that π
1
and π
2
are regular then Theorem 2.1 implies that the
right hand side of the latter equality belongs to X
∗
; thus D
∗∗
(A
∗∗
) ⊆ X
∗
, or equivalently,
D is weakly compact. Furthermore, in the same situation the latter identity together with
the equalities π
∗∗∗∗
1
= π
r∗∗∗r∗
1
and π
r∗∗∗∗r
2
= π
∗∗∗r∗r
2
imply that ,
D
∗∗
(a
∗∗
) = π
∗∗∗∗
1
(x
∗
, a
∗∗
) − π
∗∗∗r∗r
2
(a
∗∗
, x
∗
)
14 S. MOHAMMADZADEH AND H. R. E. VISHKI
and a lso
D
∗∗
(a
∗∗
) = π
r∗∗∗r∗
1
(x
∗
, a
∗∗
) − π
r∗∗∗∗r
2
(a
∗∗
, x
∗
);
which means that both D
∗∗
: (A
∗∗
, ) −→ X
∗∗∗
and D
∗∗
: (A
∗∗
, ♦) −→ X
∗∗∗
are also inner
derivations. We summarize t hese observations in the next result which is a g eneralization
of [10, Proposition 6.1].
Proposition 4.6. Let X be a A−module whose module ac tion s are regular. Then every
inner derivation D : A −→ X
∗
is weakly compact; moreover, D
∗∗
: (A
∗∗
, ) −→ X
∗∗∗
and
D
∗∗
: (A
∗∗
, ♦) −→ X
∗∗∗
are also inner derivations.
Acknowledgments
This paper was completed while the second author was visiting the University of New
South Wales in Australia. He would like to thank the School of Mathematics and Sta
tistics, especially Professor Michael Cowling for their generous hospitality. The valuable
discussion of Professor Garth Dales in his short visit to Mashhad University is also ac
knowledged.
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Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 917751159,
Mashhad, Iran
Email address: somohammadzad@yahoo.com
Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 917751159,
Mashhad, Iran
Email address: vishki@ferdowsi.um.ac.ir